The Wigner distribution function for the $\mathfrak{su}(2)$ finite oscillator and Dyck paths

TL;DR

This paper computes a discrete Wigner distribution (Wigner matrix) for the $rak{su}(2)$ finite oscillator, expressing it through sums of Krawtchouk polynomials and Dyck polynomials related to Dyck paths, revealing a simple combinatorial form.

Contribution

It introduces a novel combinatorial expression for the pre-Wigner matrix of the $rak{su}(2)$ finite oscillator using Dyck polynomials, simplifying its computation.

Findings

Derived the Wigner matrix elements using Krawtchouk polynomials.

Presented a new combinatorial expression involving Dyck polynomials.

Simplified the computation of the Wigner matrix for finite quantum systems.

Abstract

Recently, a new definition for a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum, was developed. This distribution function is defined on discrete phase-space (a finite square grid), and can thus be referred to as the Wigner matrix. In the current paper, we compute this Wigner matrix (or rather, the pre-Wigner matrix, which is related to the Wigner matrix by a simple matrix multiplication) for the case of the $\mathfrak{su}(2)$ finite oscillator. The first expression for the matrix elements involves sums over squares of Krawtchouk polynomials, and follows from standard techniques. We also manage to present a second solution, where the matrix elements are evaluations of Dyck polynomials. These Dyck polynomials are defined in terms of the well known Dyck paths. This combinatorial expression of the pre-Wigner matrix elements turns out to be particularly simple.